Theorem sqabssub 10998

Description: Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)

Assertion

Ref	Expression
sqabssub	|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )

Proof of Theorem sqabssub

Step	Hyp	Ref	Expression
1		cjsub 10834	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A - B ) ) = ( ( * ` A ) - ( * ` B ) ) )
2	1	oveq2d 5858	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) x. ( * ` ( A - B ) ) ) = ( ( A - B ) x. ( ( * ` A ) - ( * ` B ) ) ) )
3		cjcl 10790	. . . . 5 |- ( A e. CC -> ( * ` A ) e. CC )
4		cjcl 10790	. . . . 5 |- ( B e. CC -> ( * ` B ) e. CC )
5	3, 4	anim12i 336	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) e. CC /\ ( * ` B ) e. CC ) )
6		mulsub 8299	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( * ` A ) e. CC /\ ( * ` B ) e. CC ) ) -> ( ( A - B ) x. ( ( * ` A ) - ( * ` B ) ) ) = ( ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) - ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) ) )
7	5, 6	mpdan 418	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) x. ( ( * ` A ) - ( * ` B ) ) ) = ( ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) - ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) ) )
8	2, 7	eqtrd 2198	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) x. ( * ` ( A - B ) ) ) = ( ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) - ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) ) )
9		subcl 8097	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
10		absvalsq 10995	. . 3 |- ( ( A - B ) e. CC -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( A - B ) x. ( * ` ( A - B ) ) ) )
11	9, 10	syl 14	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( A - B ) x. ( * ` ( A - B ) ) ) )
12		absvalsq 10995	. . . 4 |- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
13		absvalsq 10995	. . . . 5 |- ( B e. CC -> ( ( abs ` B ) ^ 2 ) = ( B x. ( * ` B ) ) )
14		mulcom 7882	. . . . . 6 |- ( ( B e. CC /\ ( * ` B ) e. CC ) -> ( B x. ( * ` B ) ) = ( ( * ` B ) x. B ) )
15	4, 14	mpdan 418	. . . . 5 |- ( B e. CC -> ( B x. ( * ` B ) ) = ( ( * ` B ) x. B ) )
16	13, 15	eqtrd 2198	. . . 4 |- ( B e. CC -> ( ( abs ` B ) ^ 2 ) = ( ( * ` B ) x. B ) )
17	12, 16	oveqan12d 5861	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) = ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) )
18		mulcl 7880	. . . . . 6 |- ( ( A e. CC /\ ( * ` B ) e. CC ) -> ( A x. ( * ` B ) ) e. CC )
19	4, 18	sylan2 284	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( * ` B ) ) e. CC )
20	19	addcjd 10899	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( * ` B ) ) + ( * ` ( A x. ( * ` B ) ) ) ) = ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) )
21		cjmul 10827	. . . . . . 7 |- ( ( A e. CC /\ ( * ` B ) e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. ( * ` ( * ` B ) ) ) )
22	4, 21	sylan2 284	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. ( * ` ( * ` B ) ) ) )
23		cjcj 10825	. . . . . . . 8 |- ( B e. CC -> ( * ` ( * ` B ) ) = B )
24	23	adantl 275	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( * ` B ) ) = B )
25	24	oveq2d 5858	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) x. ( * ` ( * ` B ) ) ) = ( ( * ` A ) x. B ) )
26	22, 25	eqtrd 2198	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. B ) )
27	26	oveq2d 5858	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( * ` B ) ) + ( * ` ( A x. ( * ` B ) ) ) ) = ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) )
28	20, 27	eqtr3d 2200	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) = ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) )
29	17, 28	oveq12d 5860	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) = ( ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) - ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) ) )
30	8, 11, 29	3eqtr4d 2208	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   ` cfv 5188  (class class class)co 5842   CCcc 7751   + caddc 7756   x. cmul 7758   - cmin 8069   2c2 8908   ^cexp 10454   *ccj 10781   Recre 10782   abscabs 10939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941

This theorem is referenced by:  sqabssubi  11095